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Hydra - Theoretical Models entnommen aus: Meinhardt, H., A model of biological pattern formation of hypostome, tentacles and foot in Hydra: how to form structures close to each other, how to form them at a distance. Developmental Biology, : p Formation of two organizing regions at opposite positions: the head and the foot system. Assumed are two activator-inhibitor systems that are coupled via the source density. The head activator increases the source density and appears preferentially in a region of high source density. The foot system has the opposite behavior. Simulations in a linear array of cells. (a) In a small field, the maxima appear at opposite poles. Note that the foot activator appears initially closer to the head maximum but becomes shifted away due to the increasing steepness of the source density gradient. (b) If a jump in the source density exists, both activators appear close to this jump at the corresponding sides. Such a situation can be created in transplantation experiments. (c) In a large field, maxima of the head (black) and foot activator (gray) appear at several positions. Like maxima maintain a distance from one another (a general property of activator-inhibitor systems) while unlike maxima maintain a certain minimum distance because they require high or low source density (light gray), respectively. isotropes System am Anfang -> durch lokale Fluktuationen ausgelöst und durch diffusives Aktivator-Inhibitor System für Kopf und Fuss stabilisiert bilden sich Kopf und Fuß an getrennten Enden Transplantationsexperiment: Zusammensetzung zweier Stücke mit unterschiedlicher Quelldichte -> Polarität bleibt erhalten, Kopf und Fuß bilden sich wieder an entgegengesetzen Enden lange Systeme -> mehrere Maxima (Köpfe); Quelldichte Kopf Fuß

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Hydra - Theoretical Models taken from: Meinhardt, H., Wie Schnecken sich in Schale werfen. 1997, Berlin: Springer Verlag. Regeneration with maintained polarity. (a) After cutting, fragments of Hydra regenerate. The original apical-basal polarity is maintained. (b) Model: The abscissa gives, in percent of the full length, the position along the body axis. The inhibitor is assumed to have a feedback on the source density b (dashed curve) which describes the general ability of the cells to perform the autocatalysis. This source density, having a long time constant, does not change very much during regeneration of the activator-inhibitor pattern. Regions closer to the original head have an advantage in the competition for head formation, and the new maximum of the activator a (solid curve) is reliably triggered in the region that was originally closest to the apical side. dicing Formation of new steady state head activator source density

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Hydra - Theoretical Models Head and foot formation at the graft junction of mirror-image transplants. (A) Double head and (B) double foot transplants obtained by rejoining parts of adult animals sectioned at about the middle of their body column. Double head transplants form a foot, double foot transplants form a head. In both simulations (C, D) the positional value is elevated from that present in the budding region to a higher one. (C) The simulation represents the double head transplant: In the centre the level of inhibitor C is increased due to import from the surroundings. In the simulation this is achieved by decreasing the removal rate of inhibitor C. (An import of inhibitor B from both apical ends is ignored. It is obvious that in case the concentration of B is too high, autocatalysis will not start.) (D) In order to simulate the double foot transplants, the concentration of inhibitor C has to be lowered in the centre. This is achieved by increasing the removal rate of C, r c. taken from: Berking, S., A model for budding in hydra: pattern formation in concentric rings. Journal of Theoretical Biology, (1): p positional value Modell beschreibt Experimente korrekt in denen zwei Tiere in der Mitte auseinander geschnitten und dann jeweils die unteren und oberen Teile zusammengesetzt werden: (A) Positionswert sinkt an der Schnittstelle -> Fußbildung (B) Positionswert steigt an der Schnittstelle -> Kopfbildung